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Theorem grothpwex 8704
Description: Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 8700. Note that ax-pow 4379 is not used by the proof. Use axpweq 4378 to obtain ax-pow 4379. (Contributed by Gérard Lang, 22-Jun-2009.)
Assertion
Ref Expression
grothpwex  |-  ~P x  e.  _V

Proof of Theorem grothpwex
Dummy variables  y 
z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axgroth5 8701 . 2  |-  E. y
( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  /\  A. z  e.  ~P  y
( z  ~~  y  \/  z  e.  y
) )
2 simpl 445 . . . . . . . 8  |-  ( ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w
)  ->  ~P z  C_  y )
32ralimi 2783 . . . . . . 7  |-  ( A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  ->  A. z  e.  y  ~P z  C_  y
)
4 pweq 3804 . . . . . . . . 9  |-  ( z  =  x  ->  ~P z  =  ~P x
)
54sseq1d 3377 . . . . . . . 8  |-  ( z  =  x  ->  ( ~P z  C_  y  <->  ~P x  C_  y ) )
65rspccv 3051 . . . . . . 7  |-  ( A. z  e.  y  ~P z  C_  y  ->  (
x  e.  y  ->  ~P x  C_  y ) )
73, 6syl 16 . . . . . 6  |-  ( A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  ->  ( x  e.  y  ->  ~P x  C_  y ) )
87anim2i 554 . . . . 5  |-  ( ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w
) )  ->  (
x  e.  y  /\  ( x  e.  y  ->  ~P x  C_  y
) ) )
983adant3 978 . . . 4  |-  ( ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w
)  /\  A. z  e.  ~P  y ( z 
~~  y  \/  z  e.  y ) )  -> 
( x  e.  y  /\  ( x  e.  y  ->  ~P x  C_  y ) ) )
10 pm3.35 572 . . . 4  |-  ( ( x  e.  y  /\  ( x  e.  y  ->  ~P x  C_  y
) )  ->  ~P x  C_  y )
11 vex 2961 . . . . 5  |-  y  e. 
_V
1211ssex 4349 . . . 4  |-  ( ~P x  C_  y  ->  ~P x  e.  _V )
139, 10, 123syl 19 . . 3  |-  ( ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w
)  /\  A. z  e.  ~P  y ( z 
~~  y  \/  z  e.  y ) )  ->  ~P x  e.  _V )
1413exlimiv 1645 . 2  |-  ( E. y ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  /\  A. z  e.  ~P  y
( z  ~~  y  \/  z  e.  y
) )  ->  ~P x  e.  _V )
151, 14ax-mp 8 1  |-  ~P x  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    /\ w3a 937   E.wex 1551    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   class class class wbr 4214    ~~ cen 7108
This theorem is referenced by:  isrnsigaOLD  24497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-groth 8700
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-in 3329  df-ss 3336  df-pw 3803
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